Properties

Degree 2
Conductor $ 3 \cdot 17 \cdot 157 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.576·2-s + 3-s − 1.66·4-s + 0.828·5-s − 0.576·6-s − 0.479·7-s + 2.11·8-s + 9-s − 0.477·10-s + 5.91·11-s − 1.66·12-s − 2.47·13-s + 0.276·14-s + 0.828·15-s + 2.11·16-s − 17-s − 0.576·18-s + 2.75·19-s − 1.38·20-s − 0.479·21-s − 3.41·22-s + 4.50·23-s + 2.11·24-s − 4.31·25-s + 1.42·26-s + 27-s + 0.800·28-s + ⋯
L(s)  = 1  − 0.407·2-s + 0.577·3-s − 0.833·4-s + 0.370·5-s − 0.235·6-s − 0.181·7-s + 0.747·8-s + 0.333·9-s − 0.151·10-s + 1.78·11-s − 0.481·12-s − 0.685·13-s + 0.0739·14-s + 0.213·15-s + 0.529·16-s − 0.242·17-s − 0.135·18-s + 0.632·19-s − 0.308·20-s − 0.104·21-s − 0.727·22-s + 0.938·23-s + 0.431·24-s − 0.862·25-s + 0.279·26-s + 0.192·27-s + 0.151·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(8007\)    =    \(3 \cdot 17 \cdot 157\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{8007} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 8007,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;17,\;157\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;17,\;157\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 - T \)
17 \( 1 + T \)
157 \( 1 - T \)
good2 \( 1 + 0.576T + 2T^{2} \)
5 \( 1 - 0.828T + 5T^{2} \)
7 \( 1 + 0.479T + 7T^{2} \)
11 \( 1 - 5.91T + 11T^{2} \)
13 \( 1 + 2.47T + 13T^{2} \)
19 \( 1 - 2.75T + 19T^{2} \)
23 \( 1 - 4.50T + 23T^{2} \)
29 \( 1 + 6.59T + 29T^{2} \)
31 \( 1 + 2.35T + 31T^{2} \)
37 \( 1 + 9.35T + 37T^{2} \)
41 \( 1 + 7.84T + 41T^{2} \)
43 \( 1 + 10.6T + 43T^{2} \)
47 \( 1 + 10.0T + 47T^{2} \)
53 \( 1 - 10.7T + 53T^{2} \)
59 \( 1 - 0.266T + 59T^{2} \)
61 \( 1 + 7.69T + 61T^{2} \)
67 \( 1 + 12.4T + 67T^{2} \)
71 \( 1 - 7.86T + 71T^{2} \)
73 \( 1 - 2.29T + 73T^{2} \)
79 \( 1 + 6.28T + 79T^{2} \)
83 \( 1 - 11.7T + 83T^{2} \)
89 \( 1 - 10.1T + 89T^{2} \)
97 \( 1 - 0.346T + 97T^{2} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−7.50371221642112409278849739183, −6.96908254734011002889987675064, −6.23862505868735008099940394419, −5.19906465776351879925938887470, −4.73271291464622119184869751227, −3.59466139311433783535794793440, −3.46758971190567589640420367479, −1.91978134252664636310480312790, −1.39017647734735117037451265304, 0, 1.39017647734735117037451265304, 1.91978134252664636310480312790, 3.46758971190567589640420367479, 3.59466139311433783535794793440, 4.73271291464622119184869751227, 5.19906465776351879925938887470, 6.23862505868735008099940394419, 6.96908254734011002889987675064, 7.50371221642112409278849739183

Graph of the $Z$-function along the critical line